6 Answers
6

Interrogatives (who?), imperatives (stop), declaratives (me), locatives (here), and nominatives (Jane) all allow for single-word statements, as do adjectives, adverbs and so on. You'd be hard-pressed to find a category of words that are not amenable to the possibility.

In general, in almost all things pertaining to our language, you will discover that everything your English teacher taught you is wrong. See Theodore Bernstein's Miss Thistlebottom's Hobgoblins for extended commentary on this point.

Your English teacher is probably correct in the sense that those are the only complete, grammatically-correct sentences that contain only one word. But in common writing we often use one-word statements that are not complete sentences. Like Robusto's example, "Where are you going?" "Home." "Home." is not really a "sentence" as it contains neither a subject nor a verb but just an object. It conveys no clear meaning by itself, but only when heard with the preceding question. But I wouldn't be afraid to use such statements. They're perfectly acceptable to all but the most annoying pedants.

When I was in school, teachers often insisted that on a test, all answers must be complete sentences. So if the question was, say, "What is the capital of France?", a student who wrote "Paris" would be marked wrong. The student was required to write, "The capital of France is Paris." Which always struck me as rather silly: the question is right there, the teacher and I both know what it is, why do I need to repeat it? I recall my chemistry teacher once saying that on his tests it was NOT necessary to do this. That if he asked, "What chemical reaction occurs when you mix NaCl and AgNO3?", you should just write the resultant chemicals, it was not necessary to write, "Yes indeed, a chemical reaction occurs when you mix NaCl and AgNO3. ..."

The single word sentences that reply to a question are not valid sentences in written English. They might be valid in a spoken conversation but in written English (outside of direct speech) they are elliptical. A valid sentence (as opposed to a spoken utterance) requires, at the very least a subject and a verb (e.g. 'Birds fly.'). In the creation of a grammatical correct written sentence, a minimum of two words are necessary.

What (and why) does being elliptical invalidate 'outside of direct speech' but not in spoken English? I'd agree that a sentence is defined as containing a subject and a main verb at the bare minimum, but 'sentence substitutes', while obviously not being sentences, are widely accepted as not 'being ungrammatical'. See grammar.about.com/od/rs/g/sentfragterm.htm : Though in traditional grammar sentence fragments are usually treated as grammatical errors, they are commonly used by professional writers to create emphasis or particular stylistic effects , and Nordquist's 'Crots' link.
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Edwin AshworthJan 23 '13 at 15:32

This answer is circular reasoning. Of course a one-word sentence is not a sentence if you define a sentence as something that cannot be a single word. By way of exaggeration, I might as well define a sentence as something that includes a 24-letter loanword from Russian, in which case nothing on this page is a sentence. But that's pointless. The question is not what definition you can make up, but how useful it is, who does use it at all, and what happens to all the other stuff that doesn't meet it. (If "Help!" is not a sentence, what is it?)
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RegDwigнt♦Jan 23 '13 at 16:21